They mean the same thing, however 'anytime' is considered a casualism, so you might consider to use your spacebar in formal letters (any time) any_time has an extra meaning: [ any time ] any time period - Do you want any time allotted to you? [ anytime ] at any time whatever - She went there anytime she wanted to. Note, in the 2nd sentence you can replace anytime with any_time and at_any_time if you want.